[inject templates/en.xd]
[gen-title Math Notation]
[p Here are some of my controversial ideas math notation. Some of them are quite common, while others are my invention and objected to every time I bring them up.]
[section Square root;
  [$$ \sqrt{\text{This is how you write a square root.}}]
  [$$ \sqrt{\text{It has no hook at the end.}}]
]
[section Tau;
  [link [$ \tau] is good.; https://tauday.com/tau-manifesto] Use it.
]
[section Function exponentiation;
  I prefer to avoid using the notation [$ \sin^{-1}(x)], since it's unclear if it refers to [$ \arcsin(x)] or [$ [/ \sin(x)]]. Similarly, I like to write [$ \sin(x)^2], which is completely inambiguous, rather than [$ \sin^2(x)].
]
[section Factorial of a real number;
  There's no reason why the notation [$ x!] couldn't be used for the factorial of a real number, which many people awkwardly write as [$ \Gamma(x + 1)].
]
[section Logarithm;
  The notation [$ \log(x)] should not be used since nobody agrees if it's base [$ e], [$ 10] or [$ 2]. Use [$ \ln], [$ \log_{10}] or [$ \lg] respectively.
]
[section Vertical parentheses;
  Vertical parentheses are useful to apply a diacritic to a complex expression. For example, [$ [dv t][pdv L; \dot q_j]] can be succintly written as [$ \dot[v. [pdv L; \dot q_j]]].
]
[section Extracting the limit;
  When dealing with limits, it often gets repetitive to write [$ \lim] at the beginning. For example:
  [$$ [lim][/ n+1; n+2] = [lim][/ n [. 1 + [/ 1;n]]; n [. 1 + [/ 2;n]]] = [lim][/ 1 + [/ 1;n]; 1 + [/ 2;n]] = [/ 1+0; 1+0] = 1]
  In this case, we can [" extract the limit]:
  [$$ [lim][. [/ n+1; n+2] = [/ n [. 1 + [/ 1;n]]; n [. 1 + [/ 2;n]]] = [/ 1 + [/ 1;n]; 1 + [/ 2;n]]] = [/ 1+0; 1+0] = 1]
  This can also apply to other things.
]
[section Existence;
  The notation [$ \exists_n] means [" there exist exactly [$ n]]. For example, [$ \exists_1] is equivalent to (and harder to overlook than) [$ \exists!], [$ \exists_\infty] means [" there exist infinitely many], and [$ \exists_0] is the same as [$ \nexists].
]
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